New York Journal of Mathematics
New York J. Math. 8 (2002) 63–83.
Asymptotic Density in Combined Number
Systems
Karen Yeats
Abstract. Necessary and sufficient conditions are found for a combination of
additive number systems and a combination of multiplicative number systems
to preserve the property that all partition sets have asymptotic density. These
results cover and extend several special cases mentioned in the literature and
give partial solutions to two problems in [6].
Contents
1.
2.
3.
Introduction
Preliminaries
The nice cases
3.1. Additive
3.2. Multiplicative
4. The general situation
4.1. Additive
4.2. Multiplicative
References
63
64
66
66
70
74
74
78
82
1. Introduction
The recent ancestry of the problems examined in this paper began in 1937 when
abstract number systems with real valued norms were introduced by Arne Beurling
[5] for the purpose of finding minimal sufficient conditions to prove prime number
theorems. This is also the subject of Paul Bateman and Harold Diamond’s article
[2]. John Knopfmacher focused on density questions for classes of well known
algebraic and topological structures in his books [12] and [13].
Received June 27, 2001.
Mathematics Subject Classification. Primary 11N45, 11P99, Secondary 05A16, 11N80.
Key words and phrases. additive number system, multiplicative number system, asymptotic
density, partition set, ratio test, regular variation.
Thanks to Stan Burris, a patient and meticulous supervisor, to the referee for his interest
and detailed comments, and to NSERC for their Undergraduate Student Research Award which
supported this research.
ISSN 1076-9803/02
63
Karen Yeats
64
Interest in number systems with the property that is central to the discussion
below, namely that all partition sets have asymptotic density, started with Kevin
Compton’s work relating combinatorics to logic [9], [10], [11]. He proves logical limit
laws on certain classes of structures using enumeration methods. Stanley Burris’s
book Number Theoretic Density and Logical Limit Laws [6] pulls the abstract number systems and the logical asymptotic combinatorics together.
In [6] Burris poses the following problem (Problem 5.20): Find necessary and
sufficient conditions on two additive number systems A1 and A2 for all partition
sets of A1 ∗A2 to have asymptotic density. He also poses the corresponding problem
for multiplicative number systems (Problem 11.25): Find necessary and sufficient
conditions on two multiplicative number systems A1 and A2 for all partition sets
of A1 ∗ A2 to have global asymptotic density.1
This paper considers these problems when A1 and A2 themselves have the property that all partition sets have asymptotic density. Elegant and popular special
cases for additive and multiplicative systems will be considered first, followed by
the general cases.
2. Preliminaries
Definition 1. N = {0, 1, 2, . . . }.
Definition 2. A number system
A = (A, P, ∗, e, )
consists of a countable free commutative monoid (A, ∗, e), where P is the nonempty
set of indecomposable elements, and a norm.
Definition 3 ([6], 2.5 and 2.7). An additive number system is a number system
for which is an additive norm, that is, a mapping from A to the nonnegative
integers such that a = 0 iff a = e, a ∗ b = a + b, and for every n ∈ N the
set {a ∈ A : a = n} is finite.
Definition 4 ([6], 8.1 and 8.2). A multiplicative number system is a number system for which is a multiplicative norm, that is, a mapping from A to the positive
integers such that a = 1 iff a = e, a∗b = a·b, and for every positive integer
n the set {a ∈ A : a = n} is finite.
Definition 5 ([6], 2.12 and 2.28). Given a number system A, for each set B ⊆ A
the (local ) counting function of B is
b(n) = {b ∈ B : b = n},
and the global counting function of B is
B(x) =
b(n).
n≤x
Notice that B(x) is nondecreasing. As special cases we have a(n) and p(n), the
(local) counting functions of A and P, respectively, and A(x), the global counting
function of A. We will refer to a(n) and p(n) as the (local) counting functions of A
and to A(x) as the global counting function of A.
1 Burris in [6] uses + for the combination of additive number systems and × for the combination
of multiplicative number systems.
Asymptotic Density in Combined Number Systems
65
Definition 6 ([6], 3.21, 9.20). For A a number system and B1 , . . . , Bk ⊆ A, let
B1 ∗ · · · ∗ Bk = {b1 ∗ · · · ∗ bk : bi ∈ Bi }.
Definition 7 ([6], 3.23, 9.22). For A a number system and B ⊆ A,
B0
Bm
= {e}
= B
· · ∗ B
∗ ·
for m > 0
m times
B≤m
B≥m
= B0 ∪ · · · ∪ Bm for m ≥ 0
=
Bn for m ≥ 0.
n≥m
Definition 8 ([6], 3.25, 9.24). Given a number system A a subset B of A is a partition set of A if B can be written in the form
B = Pγ11 ∗ · · · ∗ Pγkk
where P1 , . . . , Pk is a finite partition of the set P of indecomposables of A, and
each γi is of the form m, ≤ m, or ≥ m.
Definition 9 ([6], 4.14, 10.4). Given number systems A1 and A2 , both additive
or both multiplicative, define A1 ∗ A2 to be the number system whose underlying
monoid is the direct product of the two monoids, and if A1 and A2 are additive then
the norm is the sum of the coordinate norms, that is, (a1 , a2 ) = a1 1 + a2 2 ,
while if A1 and A2 are multiplicative then the norm is the product of the coordinate
norms, that is, (a1 , a2 ) = a1 1 · a2 2 .
Note that A1 ∗ A2 is additive if A1 and A2 are additive and is multiplicative if
A1 and A2 are multiplicative. The set of indecomposables of A1 ∗ A2 is
{(p1 , e) : p1 ∈ P1 } ∪ {(e, p2 ) : p2 ∈ P2 }.
Let ai (n) and pi (n) be the local counting functions of Ai for i = 1, 2, and let
a(n) and p(n) be the local counting functions of A1 ∗ A2 . Then
p(n)
a(n)
= p1 (n) + p2 (n)
⎧ ⎪
a1 (i)a2 (j)
⎪
⎨
i+j=n
=
⎪
a1 (i)a2 (j)
⎪
⎩
for A an additive number system
for A a multiplicative number system.
i·j=n
Lemma 10. Let A = A1 ∗A2 be a multiplicative number system and let B = B1 ×B2
where Bi ⊆ Ai . Then
B(x) =
b1 (k)B2 (x/k).
1≤k≤x
Proof. Theorem 3.10 from [1].
Lemma 11. Let A1 and A2 be number systems both additive or both multiplicative.
Then any partition set B of A1 ∗ A2 can be written in the form
B=
k
B1j × B2j
j=1
for some k ≥ 1 where the union is disjoint and each Bij is a partition set of Ai .
Karen Yeats
66
Proof. The proof is routine and thus is left to the reader to complete by using
the definitions of m, ≥ m, and ≤ m, and the fact that if B, C, and D are sets of
numbers from a number system, then (B ∪ C) ∗ D = (B ∗ D) ∪ (C ∗ D).
3. The nice cases
In this section we will deal with the most popular additive and multiplicative systems, namely the reduced additive systems and the strictly multiplicative systems.
In Section 4 we look at the general situation.
3.1. Additive. Throughout the additive subsections of this paper A, A1 , and A2
will denote additive number systems. The most desirable of the additive systems
are those which are reduced.
Definition 12 ([6], 2.41). For f : N → N the support of f is
supp f (n) = {n ∈ N : f (n) > 0}.
Definition 13 ([6], 2.43). A is reduced if gcd(supp p(n)) = 1.
Notice that A1 ∗ A2 may be reduced even when A1 and A2 are not. However in
this subsection we will only consider reduced systems Ai .
Definition 14 ([6], 2.12). Given A, for each B ⊆ A the generating series of B is
B(x) =
b(n)xn .
n≥0
Definition 15 ([6], 1.24 and 3.18). Given ρ ≥ 0, a real-valued function f (n) that
is eventually defined on N and eventually positive is in RTρ if
f (n − 1)
= ρ,
f (n)
that is, the radius of convergence of
f (n)xn can be found by the ratio test.
A power series is in RTρ if the sequence of coefficients is in RTρ . A is in RTρ if
a(n) ∈ RTρ , which can only occur if a(n) is eventually nonzero.
lim
n→∞
Definition 16 ([6], 3.1). For B ⊆ A, the asymptotic density δ(B) of B is as follows,
provided the limit exists:
b(n)
.
δ(B) = n→∞
lim
a(n)
a(n)=0
We will use the following notational conventions.
The radius of convergence of the generating function Ai (x) of Ai is ρi , the
counting functions are ai (n) and pi (n), and when defined δi (Bi ) is the asymptotic
density of Bi with respect to Ai for Bi ⊆ Ai . For A the preceding apply with the
removal of the subscript i.
Since P, P1 , and P2 are nonempty by definition, we know that ρ, ρ1 , and ρ2 are
in [0, ∞). From Lemma 2.23 of [6] we know that ρ, ρ1 , and ρ2 are in [0, 1].
Lemma 17. Let A = A1 ∗ A2 and B = B1 × B2 where Bi ⊆ Ai . Then:
1. B(x)
B2 (x).
= B1 (x) ·
2.
B(n)xn =
B1 (n)xn · B2 (x).
n≥0
n≥0
Asymptotic Density in Combined Number Systems
67
Proof. Both
items follow from Proposition 3.33 of [6]. For the second item notice
that B(n) = k≤n B1 (k)b2 (n − k) is the same relation as that which holds between
b(n), b1 (n), and b2 (n).
Lemma 18. If A = A1 ∗ A2 then ρ = min{ρ1 , ρ2 }.
Proof. ρ = min{ρ1 , ρ2 } by Lemma 17 since A1 (x) and A2 (x) are power series with
nonnegative coefficients.
Definition 19. A has Property I if all partition sets of A have asymptotic density.
A has Property II if ρ > 0, A(ρ) = ∞, and a(n) ∈ RTρ .
Our goal is to analyse when Property I holds. Some results of Bell, Bell et al.,
Burris and Sárközy, and Warlimont follow; then we will proceed towards the main
result of the additive subsection, Theorem 28.
Proposition 20 ([6], 3.28). If A has Property I then δ(P) = 0.
Proposition 21 ([3], [15], [4]). If δ(P) = 0 then ρ > 0 and A(ρ) = ∞.
Proposition 22 ([6], 3.30). A has Property I and is reduced implies a(n) ∈ RTρ .
Corollary 23. A has Property I and is reduced implies A has Property II.
Proposition 24. Let A1 and A2 have Property II and be reduced, and let A =
A1 ∗ A2 . If ρ1 ≤ ρ2 , then:
0
if ρ1 = ρ2
a1 (n)
• lim
=
n→∞ a(n)
A2 (ρ1 )−1 if ρ1 < ρ2 .
a2 (n)
= 0.
• lim
n→∞ a(n)
Proof. For the second item take n large enough that a2 (n) > 0. Then for m < n
n
m
a2 (n − k)
a2 (n − k)
a(n)
=
≥
.
a1 (k)
a1 (k)
a2 (n)
a2 (n)
a2 (n)
k=0
k=0
So
lim inf
n→∞
m
a(n)
≥
a1 (k)ρk2 .
a2 (n)
k=0
Now A1 (ρ2 ) = ∞; so by taking the limit as m → ∞ we get the desired result.
For the first item if ρ1 = ρ2 apply the above proof with 1 and 2 interchanged; if
ρ1 < ρ2 then apply Schur’s Theorem.2
2 By
this is meant Schur’s Tauberian Theorem which can be found in [6], 3.42. It states: Let
S(x) and T(x) be two power series such that, for some ρ ≥ 0, T(x) ∈ RTρ , and S(x) has radius
of convergence greater than ρ. Then
n x
S(x) · T(x)
lim
= S(ρ).
n→∞
xn T(x)
If S(ρ) > 0 then this can be expressed as xn S(x) · T(x) ∼ S(ρ) · xn T(x).
Karen Yeats
68
Proposition 25. Let A1 and A2 have Property I and be reduced with ρ1 = ρ2 .
Then A1 ∗ A2 has Property I and for B a partition set of A1 ∗ A2 and Bij as in
Lemma 11 we get
δ(B) =
k
δ1 (B1j )δ2 (B2j ).
j=1
Proof. Let A = A1 ∗ A2 .
First, consider partition sets of A of the form B = B1 × B2 where each Bi is a
partition set of Ai .
From bi (n)/ai (n) → δi (Bi ) it follows that for all > 0 there is an N such that
for m, n ≥ N ,
b1 (m)b2 (n) − δ1 (B1 )δ2 (B2 )a1 (m)a2 (n) ≤ a1 (m)a2 (n).
o
a(n)
Then,
using
Proposition
24,
and
the
fact
that
a
linear
combination
r
i
is o a(n) , we have
b(n) − δ1 (B1 )δ2 (B2 )a(n)
n
= b1 (j)b2 (n − j) − δ1 (B1 )δ2 (B2 )a1 (j)a2 (n − j)
j=0
n−N
≤ b1 (j)b2 (n − j) − δ1 (B1 )δ2 (B2 )a1 (j)a2 (n − j)
j=N
−1
N
+
b1 (j)b2 (n − j) − δ1 (B1 )δ2 (B2 )a1 (j)a2 (n − j)
j=0
−1
N
+
b1 (n − j)b2 (j) − δ1 (B1 )δ2 (B2 )a1 (n − j)a2 (j)
j=0
−1
N
≤ a(n) + b1 (j)o a(n) − δ1 (B1 )δ2 (B2 )a1 (j)o a(n) j=0
−1
N
+
o a(n) b2 (j) − δ1 (B1 )δ2 (B2 )o a(n) a2 (j)
j=0
= a(n) + o a(n) .
Therefore δ(B) = δ1 (B1 )δ2 (B2 ).
Thus for a general partition set B of A
δ(B) =
k
δ1 (B1j )δ2 (B2j )
j=1
by Lemma 11 and Lemma 3.2 from [6]. Therefore A has Property I.
Lemma 26. If lim c(n)/d(n) = > 0 and d(n) ∈ RTρ then c(n) ∈ RTρ .
n→∞
Asymptotic Density in Combined Number Systems
69
Proof. The hypotheses imply c(n) is eventually nonzero, and then
c(n − 1)
c(n − 1) d(n) d(n − 1)
= lim
= ρ.
n→∞
n→∞ d(n − 1) c(n)
c(n)
d(n)
lim
Proposition 27. Let A1 and A2 have Property I and be reduced with ρ1 < ρ2 .
Then A1 ∗ A2 has Property I. Let B be a partition set of A1 ∗ A2 and let Bij be as
in Lemma 11. Then
δ(B) =
k
δ1 (B1j )
j=1
B2j (ρ1 )
.
A2 (ρ1 )
Proof. Let A = A1 ∗ A2 . We know A1 ∈ RTρ by Proposition 22. Thus A ∈ RTρ
by Proposition 3.44 of [6].
Consider partition sets of A of the form B = B1 × B2 where Bi is a partition set
of Ai , i = 1, 2.
First, suppose δ1 (B1 ) = 0. Then B1 (x) ∈ RTρ by Lemma 26. Thus by Lemma 17
we can apply Schur’s Theorem to get b(n) ∼ b1 (n)B2 (ρ1 ) and a(n) ∼ a1 (n)A2 (ρ1 ).
Hence
δ(B) = δ1 (B1 )
B2 (ρ1 )
.
A2 (ρ1 )
Second, suppose δ1 (B1 ) = 0; then for all > 0 there exists an N such that
b1 (n)/a1 (n) < for n ≥ N . Thus, since b2 (n) ≤ a2 (n) for all n,
b(n)
=
n
b1 (n − j)b2 (j)
j=0
≤
n−N
b1 (n − j)a2 (j) +
j=0
≤ n−N
N
−1
b1 (j)o a(n)
by Proposition 24
j=0
a1 (n − j)a2 (j) + o a(n)
j=0
≤ a(n) + o a(n)
which gives δ(B) = 0. Therefore in both cases δ(B) = δ1 (B1 )
Hence for a general partition set B of A
δ(B) =
k
j=1
δ1 (B1j )
B2 (ρ1 )
.
A2 (ρ1 )
B2j (ρ1 )
A2 (ρ1 )
by Lemma 11 and Lemma 3.2 from [6]. Therefore A has Property I.
Theorem 28. Let A1 and A2 have Property I and be reduced with ρ1 ≤ ρ2 . Then
A1 ∗ A2 has Property I and for B a partition set of A1 ∗ A2 and Bij as in Lemma 11
Karen Yeats
70
we get
δ(B) =
⎧
k
⎪
⎪
B2j (ρ1 )
⎪
⎪
δ1 (B1j )
⎪
⎨
A2 (ρ1 )
j=1
k
⎪
⎪
⎪
⎪
⎪
⎩
δ1 (B1j )δ2 (B2j )
if ρ1 < ρ2 ,
if ρ1 = ρ2 .
j=1
Proof. If ρ1 = ρ2 apply Proposition 25; otherwise apply Proposition 27.
Remark 29. If we assume only that A1 ∗ A2 has Property I then we can not
conclude that A1 and A2 have Property I. For instance if 1 < q1 < q2 , where q2 is an
integer, and we take A1 to be the additive number system with p1 (n) = q1n /n2 and
A2 to be the additive number system with p2 (n) = q2n , then p(n) = q2n + q1n /n2 =
q2n + O(q1n ); so by Theorem 5.17 of [6], based on the asymptotics of Knopfmacher,
Knopfmacher, and Warlimont, A1 ∗ A2 has Property I. However ρ1 = 1/q1 and
P1 (1/q1 ) < ∞ which gives A1 (1/q1 ) < ∞; so A1 does not have Property I.3
3.2. Multiplicative. Throughout the multiplicative subsections of this paper, unless otherwise specified, A, A1 , and A2 will denote multiplicative number systems.
Similarly to the additive situation, the most desirable multiplicative systems are
those which are strictly multiplicative.
Definition 30 ([6], 9.39 and 9.48). A is discrete if there is a positive integer λ
such that a is an integer power of λ for any a ∈ A. A is strictly multiplicative if
it is not discrete.
Strictly multiplicative systems will be the focus of this subsection.
Definition 31 ([6], 8.6). Given A, for each set B ⊆ A the generating series of B
is the Dirichlet series:
b(n)n−x .
B(x) =
n≥1
Definition 32 ([6], 7.19 and 9.16). Given α ∈ R, a function f (x) that is eventually
defined on R, and eventually positive, is in RVα if for all y > 0
f (xy)
lim
= yα ,
x→∞ f (x)
that is, f (x) has regular variation at infinity of index α. A is in RVα if A(x) ∈ RVα .
Definition 33 ([6], 9.3). For B ⊆ A, the global asymptotic density Δ(B) of B is as
follows, provided the limit exists:
B(n)
.
Δ(B) = lim
n→∞ A(n)
We will use the following notational conventions.
The abscissa of convergence of the generating function Ai (x) of Ai is αi , the
functions ai (n), pi (n), and Ai (x) are the local and global counting functions, and
when defined Δi (Bi ) is the global asymptotic density of Bi with respect to Ai for
Bi ⊆ Ai . For A the preceding apply with the removal of the subscript i.
3 This
counterexample is inspired by one suggested by the referee.
Asymptotic Density in Combined Number Systems
71
Since P, P1 , and P2 are nonempty by definition, we know that α, α1 , and α2 are
nonnegative.
Lemma 34. Let A = A1 ∗ A2 and B = B1 × B2 where Bi ⊆ Ai . Then
B(x) = B1 (x) · B2 (x).
Proof. Apply Proposition 9.30 from [6].
Lemma 35. If A = A1 ∗ A2 then α = max{α1 , α2 }.
Proof. This follows from Lemma 34 since A1 (x) and A2 (x) both have nonnegative
coefficients.
Definition 36. A has Property I means that all partition sets of A have global
asymptotic density. A has Property II if α < ∞, A(α) = ∞, and A ∈ RVα .
As in the additive case Property I is the central property of interest. Some important results of Burris and Sárközy, and Warlimont follow; then we will proceed towards the main result of the multiplicative subsection, Theorem 47. Proposition 39
is a recently announced result of Warlimont proving conjecture 9.70 from [6].
Proposition 37 ([6], 9.28). If A has Property I then Δ(P) = 0.
Proposition 38 ([14]). If Δ(P) = 0 then α < ∞.
Proposition 39 ([16]). If Δ(P) = 0 then A(α) = ∞.
Proposition 40 ([6], 9.50). If A is strictly multiplicative and has Property I then
A ∈ RVα .
Corollary 41. A has Property I and is strictly multiplicative implies A has Property II.
We also need a multiplicative analogue of Schur’s Theorem:
Proposition 42 ([6, 9.53]). Given α ∈ R, suppose S(x) and T(x) are two Dirichlet
series such that T(x) has nonnegative coefficients, T (x) ∈ RVα , and the abscissa
of absolute convergence of S(x) is less than α. Let R(x) = S(x) · T(x). Then
lim
x→∞
R(x)
= S(α).
T (x)
If S(α) > 0 then this can be expressed as R(x) ∼ S(α) · T (x).
Proposition 43. Let A1 and A2 have Property II and be strictly multiplicative,
and let A = A1 ∗ A2 . If α1 ≥ α2 , then:
0
if α1 = α2
A1 (n)
• lim
=
n→∞ A(n)
A2 (α1 )−1 if α1 > α2 .
A2 (n)
= 0.
• lim
n→∞ A(n)
Proof.
A(n)
A2 (n/i)
A2 (n/i)
≥
a1 (i)
a1 (i)
=
A2 (n)
A2 (n)
A2 (n)
i≤n
i≤m
Karen Yeats
72
for fixed m < n. So
lim inf
n→∞
A(n)
a1 (i)i−α2 .
≥
A2 (n)
i≤m
Now A1 (α2 ) = ∞; so by taking the limit as m → ∞ we get the second item.
For the first item, if α1 = α2 apply the above proof with 1 and 2 interchanged;
if α1 > α2 then apply Proposition 42.
Proposition 44. Let A1 and A2 have Property I and be strictly multiplicative with
α1 = α2 . Then A1 ∗ A2 has Property I. Let B be a partition set of A1 ∗ A2 and let
Bij be as in Lemma 11. Then
Δ(B) =
k
Δ1 (B1j )Δ2 (B2j ).
j=1
Proof. Let A = A1 ∗ A2 .
First, consider partition sets of A of the form B = B1 × B2 where each Bi is a
partition set of Ai .
Take an > 0. Let N be such that Bi (n) − Δi (Bi )Ai (n) < Ai (n) for n ≥ N
and i = 1, 2. Notice that
(1)
n
b1 (j)A2 (n/j) =
j=1
n
B1 (n/j)a2 (j),
j=1
since both sides are equal to the global counting function of B1 × A2 . Then
B(n) − Δ1 (B1 )Δ2 (B2 )A(n)
n
= b1 (j)B2 (n/j) − Δ1 (B1 )Δ2 (B2 )a1 (j)A2 (n/j)
j=1
n
≤ b1 (j)B2 (n/j) − Δ2 (B2 )b1 (j)A2 (n/j)
j=1
n
+
Δ2 (B2 )b1 (j)A2 (n/j) − Δ1 (B1 )Δ2 (B2 )a1 (j)A2 (n/j)
j=1
n
= b1 (j)B2 (n/j) − Δ2 (B2 )b1 (j)A2 (n/j)
j=1
I
n
+ Δ2 (B2 ) B1 (n/j)a2 (j) − Δ1 (B1 )A1 (n/j)a2 (j) by (1).
j=1
II
Applying the triangle equality gives
Asymptotic Density in Combined Number Systems
Term I ≤
b1 (j)B2 (n/j) − Δ2 (B2 )A2 (n/j)
1≤j≤n/N
+
73
b1 (j)B2 (n/j) − Δ2 (B2 )A2 (n/j)
n/N <j≤n
≤ b1 (j)A2 (n/j)
1≤j≤n/N
+
1≤k<N
n
n
k+1 <j≤ k
≤ A(n) +
N
−1
B1
b1 (j)B2 (k) − Δ2 (B2 )A2 (k)
n
k
− B1
n
k+1
B2 (k) − Δ2 (B2 )A2 (k)
k=1
= A(n) + o A(n)
by Proposition 43.
Term II can be treated in the same way. Therefore Δ(B) = Δ1 (B1 )Δ2 (B2 ).
Thus for a general partition set B of A
Δ(B) =
k
Δ1 (B1j )Δ2 (B2j )
j=1
by Lemma 11 and Lemma 9.4 from [6]. Therefore A has Property I.
Lemma 45. Let B be a partition set of A. If Δ(B) = 0 and A(x) ∈ RVα then
B(x) ∈ RVα .
Proof. B(x) is eventually nonzero, since Δ(B) = 0. Then, for all y > 0,
lim
x→∞
B(xy)
B(x)
B(xy) A(x) A(xy)
A(xy) B(x) A(x)
B(xy) A(x) A(xy)
= lim
= yα .
x→∞ A(xy) B(x) A(x)
=
lim
x→∞
Proposition 46. Let A1 and A2 have Property I and be strictly multiplicative with
α1 > α2 . Then A1 ∗ A2 has Property I. Also for B a partition set of A1 ∗ A2 and
for Bij as in Lemma 11 we get
Δ(B) =
k
j=1
Δ1 (B1j )
B2j (α1 )
.
A2 (α1 )
Proof. Let A = A1 ∗ A2 .
Consider partition sets of A of the form B = B1 × B2 where Bi is a partition set
of Ai , i = 1, 2.
First, suppose Δ1 (B1 ) = 0. Then B1 (x) ∈ RVα1 by Lemma 45. Thus by
Lemma 34 we can apply Proposition 42 to get B(x) ∼ B1 (x)B2 (α1 ) and A(x) ∼
A1 (x)A2 (α1 ). So, as n → ∞,
B(n)
B2 (α1 )
→ Δ1 (B1 )
.
A(n)
A2 (α1 )
Karen Yeats
74
Second, suppose Δ1 (B1 ) = 0. Then for all > 0 there is an N such that
B1 (n)/A1 (n) < for n ≥ N . Thus
B(n) =
B1 (n/j)b2 (j) +
B1 (n/j)b2 (j)
1≤j≤n/N
≤
n/N <j≤n
1≤j≤n/N
≤ 1≤k<N
n
n
k+1 <j≤ k
A1 (n/j)a2 (j) +
1≤j≤n/N
≤ A(n) +
B1 (n/j)b2 (j) +
B1 (k)b2 (j)
n B1 (k) B2 nk − B2 k+1
1≤k<N
B1 (k)o A(n)
by Proposition 43
1≤k<N
≤ A(n) + o A(n)
which gives Δ(B) = 0. Therefore in both cases Δ(B) = Δ1 (B1 )
Hence for a general partition set B of A
Δ(B) =
k
j=1
Δ1 (B1j )
B2 (α1 )
.
A2 (α1 )
B2j (α1 )
A2 (α1 )
by Lemma 11 and Lemma 9.4 from [6]. Therefore A has Property I.
Theorem 47. Let A1 and A2 have Property I and be strictly multiplicative with
α1 ≥ α2 . Then A = A1 ∗ A2 has Property I. Also for B a partition set of A1 ∗ A2
and for Bij as in Lemma 11 we get
⎧
k
⎪
⎪
B2j (α1 )
⎪
⎪
if α1 > α2 ,
Δ1 (B1j )
⎪
⎨
A2 (α1 )
j=1
Δ(B) =
k
⎪
⎪
⎪
⎪
Δ1 (B1j )Δ2 (B2j ) if α1 = α2 .
⎪
⎩
j=1
Proof. If α1 = α2 apply Proposition 44; otherwise apply Proposition 46.
4. The general situation
4.1. Additive. Now, returning to the additive notational conventions, we extend
the results to the case when A1 and A2 are not necessarily reduced.
Definition 48. Let d = gcd(supp p(n)) and let j|d. Define A∗j to be the additive
number system obtained from A by altering the norm so that a∗j = a/j. Let
a∗j (n) and p∗j (n) be the counting functions of A∗j , and let ρ∗j be the radius of
convergence of A∗j (x), the generating function of A∗j .
Notice that a∗j (n) = a(nj), p∗j (n) = p(nj), gcd(supp p∗j (n)) = 1, and ρ∗j = ρj .
If d = j then we drop the j in the exponent and write A∗ , which is a reduced
additive system called the reduced form of A. Notice also that if A is not reduced
then A ∈ RTρ since a(n) is infinitely often zero.
As an additional notational convention we will let di = gcd(supp pi (n)) and
d = gcd(supp p(n)).
Asymptotic Density in Combined Number Systems
75
Here is an example of the value of working with nonreduced systems. For q
a power of a prime and d a positive integer, define the additive number system
Aq,d as the system formed from the monic elements of Fq [xd ] with polynomial
multiplication as the operation and g(x) = deg g(x).
Let aq,d (n) be the local counting function for Aq,d . Then
q m if dm = n
aq,d (n) =
0
if d n.
So the radius of convergence of the generating function of Aq,d is
ρ=
1
q 1/d
.
Now, for all d, A∗q,d looks like Aq,1 under the mapping g(xd ) → g(x). From aq,1 (n) =
q n we see by the Knopfmacher, Knopfmacher, and Warlimont asymptotics (see
Theorem 5.17, [6]) that Aq,1 has Property I and so, as we will see in Lemma 51,
Aq,d has Property I for all d.
Determining when Aq1 ,d1 ∗ Aq2 ,d2 has Property I is rather more involved.
Theorem 57, the main theorem of this subsection, shows that Aq1 ,d1 ∗ Aq2 ,d2 has
Property I iff
1/d
1/d
• 1/q1 1 < 1/q2 2 and d1 |d2 , or
1/d
1/d
• 1/q1 1 > 1/q2 2 and d2 |d1 , or
1/d1
1/d
• 1/q1
= 1/q2 2 .
For instance, A2,2 ∗ A2,4 and A2,1 ∗ A2,3 have Property I but A2,2 ∗ A2,3 does
not. Such facts require a careful study of nonreduced systems.
b∗ (n)
, provided δ(B) exists.
n→∞ a∗ (n)
Lemma 49. For all B ⊆ A, δ(B) = lim
Proof. From Hua’s Theorem (as found for instance in [6], 2.49), a∗ (n) is eventually
b∗ (n)
b(n)
= n→∞
lim
nonzero; so lim ∗
.
n→∞ a (n)
a(n)
a(n)=0
The next lemma follows easily from the definitions.
∗
∗
Lemma 50. Let A = A1 ∗ A2 ; then d = gcd(d1 , d2 ). Let j|d; then A∗j = A1j ∗ A2j
iff A = A1 ∗ A2 .
Lemma 51. A∗j has Property I iff A has Property I, whenever j|d.
Proof. A and A∗j have the same reduced form, so apply Lemma 3.3 from [6]. From this we get a slight variation on Proposition 22.
Proposition 52. A has Property I implies a∗ (n) ∈ RTρ∗ .
The key to the more general case is Proposition 54, a modified version of Proposition 24. In order to prove Proposition 54 we will need the following Lemma.
Lemma 53. If f (n) ∈ RTρ , ρ > 0, and n≥0 f (n)ρn = ∞ then for all k and all
m ≥ 1 we have
f (n)ρn = ∞.
n≡k mod m
Karen Yeats
76
Proof. Pick an > 0. Let N be such that f (j + 1) > 0 and f (j)/f (j + 1) < + ρ
for j > N . Then, for suitable constants C1 and C,
f (j)ρj =
f (j)ρj +
f (j)ρj
j≡k mod m
j≡k mod m
j≤N
≤ C1 + ( + ρ)
= C +
Now
j≡k mod m
j>N
f (j + 1)ρj
j≡k mod m
j>N
+ρ
ρ
f (j)ρj .
j≡k+1 mod m
f (n)ρ = ∞ implies that for all m we have
n
n≥0
f (j)ρj = ∞ for
j≡k mod m
some k. Then by the above the latter equation holds for all k
Proposition 54. Let A = A1 ∗ A2 , gcd(d1 , d2 ) = 1, A1 (ρ1 ) = ∞, and
If ρ1 ≤ ρ2 then lim a2 (n − j)/a(n) = 0 for all integers j ≥ 0.
A∗2
∈ RTρ∗2 .
n→∞
Proof. Let us restrict out attention to n large enough that a(n − j) > 0. This
is possible since d = gcd(d1 , d2 ) = 1; so A is reduced. For d2 n − j we have
a2 (n − j) = 0, and so a2 (n − j)/a(n) = 0. Assume d2 |n − j, and let d2 m = n − j.
Now
d2
m+j
a(d2 m + j)
a2 (d2 m + j − k)
=
a1 (k)
a2 (d2 m)
a2 (d2 m)
k=0
a2 (d2 m − d2 i)
=
a1 (d2 i + j)
a2 (d2 m)
−j/d2 ≤i≤m
=
a1 (d2 i + j)
−j/d2 ≤i≤m
≥
k
a1 (d2 i + j)
i=0
a∗2 (m − i)
a∗2 (m)
a∗2 (m − i)
a∗2 (m)
for fixed k < m. Thus
lim inf
m→∞
a(d2 m + j)
a2 (d2 m)
k
≥
a1 (d2 i + j) lim inf
m→∞
i=0
k
=
a∗2 (m − i)
a∗2 (m)
a1 (d2 i + j)ρ2d2 i .
i=0
Taking the limit as k → ∞, we have
lim inf
m→∞
a(d2 m + j)
a2 (d2 m)
≥
∞
i=0
= ρ−j
2
a1 (d2 i + j)ρ2d2 i
a1 (d1 )ρ2d1 d1 ≡j mod d2
≥
ρ−j
2
d1 ≡j mod d2
a∗1 ()(ρ∗1 )
Asymptotic Density in Combined Number Systems
77
since ρd21 ≥ ρd11 = ρ∗1 . Now since gcd(d1 , d
2 ) = 1, d1 ≡ j mod d2 is the same as
≡ c mod d2 for some c. We know that ≥0 a∗1 ()(ρ∗1 ) = ∞, and ρ1 > 0 since
A1 (ρ1 ) = ∞. Apply Lemma 53 to get lim a2 (n − j)/a(n) = 0.
n→∞
Proposition 55. Let A1 and A2 have Property I with ρ1 = ρ2 . Then A1 ∗ A2 has
Property I. Let B be a partition set of A1 ∗ A2 and let Bij be as in Lemma 11. Then
δ(B) =
k
δ1 (B1j )δ2 (B2j ).
j=1
Proof. Let A = A1 ∗ A2 . By Lemmas 50 and 51 we need only consider d = 1.
Since ρ1 = ρ2 we can switch the roles of A1 and A2 in Proposition 54 to get
a1 (n − j) = o a(n) for all integers j ≥ 0.
From bi (ndi )/ai (ndi ) → δi (Bi ) it follows that for all > 0 there is an N such
that for md1 , nd2 ≥ N ,
b1 (md1 )b2 (nd2 ) − δ1 (B1 )δ2 (B2 )a1 (md1 )a2 (nd2 ) < a1 (md1 )a2 (nd2 ).
Clearly ai (k) = 0 gives bi (k) = 0 for all k; so for j, k ≥ N ,
b1 (j)b2 (k) − δ1 (B1 )δ2 (B2 )a1 (j)a2 (k) ≤ a1 (j)a2 (k).
Now continue as in the proof of Proposition 25 using Proposition 54 in place of
Proposition 24.
Proposition 56. Let A1 and A2 have Property I with ρ1 < ρ2 . Then A1 ∗ A2 has
Property I iff d1 |d2 . If d1 |d2 then for B a partition set of A1 ∗ A2 and for Bij as in
Lemma 11 we get
δ(B) =
k
j=1
δ1 (B1j )
B2j (ρ1 )
.
A2 (ρ1 )
Proof. Let A = A1 ∗ A2 . d1 |d2 iff d1 = d since d = gcd(d1 , d2 ). We need
only
consider the case when d = 1, that is, A is reduced, because d1 |d2 iff dd1 dd2 and
Lemmas 50 and 51 give A1 ∗A2 has Property I iff A1∗d ∗A∗2d has Property I. A1 ∈ RTρ
iff A ∈ RTρ by Proposition 3.44 of [6].
(⇒) Suppose that A has Property I and radius of convergence ρ. Then A ∈ RTρ
since A is reduced, and so A1 ∈ RTρ . Therefore A1 is reduced, and so d1 = 1 = d.
(⇐) Suppose that d1 = 1 = d; then A1 ∈ RTρ1 . Now continue as in the proof of
Proposition 27 using Proposition 54 in place of Proposition 24.
Theorem 57. Let A1 and A2 have Property I with ρ1 ≤ ρ2 . Then A1 ∗ A2 has
Property I iff d1 |d2 or ρ1 = ρ2 . If A1 ∗ A2 has Property I then for B a partition set
of A1 ∗ A2 and Bij as in Lemma 11 we get
⎧
k
⎪
⎪
B2j (ρ1 )
⎪
⎪
δ1 (B1j )
if ρ1 < ρ2 ,
⎪
⎨
A2 (ρ1 )
j=1
δ(B) =
k
⎪
⎪
⎪
⎪
δ1 (B1j )δ2 (B2j ) if ρ1 = ρ2 .
⎪
⎩
j=1
Proof. If ρ1 = ρ2 apply Proposition 55; otherwise apply Proposition 56.
78
Karen Yeats
For any additive number system A define Aq , the extension of A by an indecomposable q, to be the additive system formed by adding a new indecomposable q to
P, by letting Aq be the set of formal expressions qm ∗ a, for m a nonnegative integer
and a ∈ A, and by extending the norm with the definition qm ∗ a = mq + a.
Corollary 58. Suppose A has Property I. An extension Aq has Property I iff dq
or ρ = 1.
Proof. We can form an additive number system Q = (AQ , PQ , ∗, e, ) generated
by q by letting PQ = {q}, AQ = {qm : m ≥ 0}, qm1 ∗ qm2 = qm1 +m2 , e = q0 , and
qm = mq. Then Aq ∼
= A ∗ Q, Q has Property I, ρQ = 1, and dQ = q. Now
apply Theorem 57.
If A is reduced in the above corollary then we have Theorem 3.59 of [6].
Corollary 59. Let A1 , . . . , An be additive number systems with A∗1 , . . . , A∗n in
RT1 . Then (A1 ∗ · · · ∗ An )∗ is also in RT1 .
Proof. Let A = A1 ∗ · · · ∗ An . It suffices to prove the theorem for n = 2. By
Theorem 4.2 of [6] A∗i ∈ RT1 iff A∗i has Property I and ρi = 1, and likewise for A∗ .
Now apply Theorem 57.
This gives the additive half of Theorem 16.1 from [7]. If all the Ai are reduced
then we have Stewart’s Theorem ([6], 4.15). We can also use Theorem 57 to prove
the following corollary, extending a result of Bateman and Erdős which in our
notation says: if p(n) ≤ 1, for n ≥ 1, then A∗ ∈ RT1 , ([6], 4.13).
Corollary 60. If p(n) ≤ c, for n ≥ 1, then A∗ ∈ RT1
Proof. Let m = max{p(n) : n ≥ 1}, which exists since p(n) ≤ c is integer valued.
Then we can construct additive number systems Ai , 1 ≤ i ≤ m, such that pi (n) ≤ 1
for 1 ≤ i ≤ m and A = A1 ∗ · · · ∗ Am . By the Bateman and Erdős result each
A∗i ∈ RT1 ; so by Corollary 59 A∗ ∈ RT1 .
4.2. Multiplicative. Now, using the multiplicative notational conventions, we
extend the results to the case when A1 and A2 may be discrete.
As an additional notational convention, if A is discrete λ will denote the largest
integer such that a is an integer power of λ for all a ∈ A; and similarly for λi
when Ai is discrete.
Although A having Property I does not imply A ∈ RVα the following lemma
gives a weaker result of a similar flavour.
Lemma 61. For any multiplicative number system A with Property I and abscissa
A(n/i)
≥ Ci−α , for all
of convergence α, there exists a C > 0 such that lim inf
n→∞
A(n)
integers i > 0.
Proof. If A is strictly multiplicative then, by Corollary 9.50 from [6], A ∈ RVα ,
and so the lemma holds with C = 1.
Suppose A is discrete multiplicative; then A = (A, P, ∗, e, logλ ) is a reduced
additive number system with all partition sets having global asymptotic density, and
with the radius of convergence of A(x)
being λ−α . Notice also that A(x)
= A(λx )
Asymptotic Density in Combined Number Systems
79
Thus, by Proposition
(Section 9.8.2 of [6] discusses relationships between A and A).
3.20 and Lemma 3.29 from [6], A(n) ∈ RTλ−α ; so
A(n/i)
A(n)
=
≥
A(log
λ (n) − logλ (i))
A(log
λ (n))
A(log
λ (n) − logλ (i))
A(logλ (n))
→ λ−αlogλ i
.
Therefore
lim inf
n→∞
A(n/i)
≥ λ−αlogλ i
≥ λ−α(logλ i+1) = λ−α i−α .
A(n)
So let C = λ−α .
Proposition 62. Let A1 and A2 have Property I and let A = A1 ∗ A2 . If α1 ≥ α2 ,
then:
0
if α1 = α2
A1 (n)
• lim
=
n→∞ A(n)
A2 (α1 )−1 if α1 > α2 .
A2 (n)
= 0.
• lim
n→∞ A(n)
Proof.
A(n)
A2 (n/i)
A2 (n/i)
=
≥
a1 (i)
a1 (i)
A2 (n)
A2 (n)
A2 (n)
i≤n
i≤m
for fixed m < n. So
A(n)
A2 (n/i)
≥ lim inf
≥ C
a1 (i)
a1 (i)i−α
lim inf
n→∞ A2 (n)
n→∞
A2 (n)
i≤m
i≤m
by Lemma 61. By taking the limit as m → ∞ we get the second item.
For the first item, if α1 = α2 apply the above proof with 1 and 2 interchanged;
if α1 > α2 then apply Proposition 42.
Proposition 63. Let A1 and A2 have Property I with α1 = α2 . Then A1 ∗ A2 has
Property I. Let B be a partition set of A1 ∗ A2 and let Bij be as in Lemma 11. Then
Δ(B) =
k
Δ1 (B1j )Δ2 (B2j ).
j=1
Proof. Follow the proof to Proposition 44 replacing references to Lemma 43 with
references to Lemma 62.
Proposition 64. Let A1 and A2 have Property I with α1 > α2 , and let A1 be
strictly multiplicative. Then A1 ∗ A2 has Property I. Let B be a partition set of
A1 ∗ A2 and let Bij be as in Lemma 11. Then
Δ(B) =
k
j=1
Δ1 (B1j )
B2j (α1 )
.
A2 (α1 )
Karen Yeats
80
Proof. Follow the proof to Proposition 46 replacing references to Lemma 43 with
references to Lemma 62.
For the purposes of the next lemma we will briefly escape from our multiplicative
notational conventions. This lemma corresponds to Proposition 3.44 from [6], but
is concerned with global counting functions rather than local counting functions.
Lemma 65. SupposeA = A1 ∗ A2 , all three of which are additive number
systems.
Additionally suppose n≥0 A1 (n)xn has radius of convergence ρ and n≥0 A2 (n)xn
has radius of convergence ρ2 > ρ. Then A(n) ∈ RTρ iff A1 (n) ∈ RTρ .
Proof. By Lemma 2.31 from [6] the radius of convergence of A2 (x) is also ρ2 . By
Corollary 2.39 from [6] we have that [1/A2 ](x) has radius of convergence at least
ρ2 , where [1/A2 ](x) is the power series expansion of 1/A2 (x). Then, by Lemma 17
A(n)xn =
A1 (n)xn · A2 (x)
n≥0
n≥0
A1 (n)xn
=
[1/A2 ](x) ·
n≥0
A(n)xn .
n≥0
It follows from Schur’s Theorem that A(n) ∈ RTρ iff A1 (n) ∈ RTρ .
Proposition 66. Let A1 and A2 have Property I with α1 > α2 , and let A1 and
A2 be discrete with λ2 a power of λ1 . Then A1 ∗ A2 has Property I. Let B be a
partition set of A1 ∗ A2 and let Bij be as in Lemma 11. Then
Δ(B) =
k
j=1
Δ1 (B1j )
B2j (α1 )
.
A2 (α1 )
Proof. Let A = A1 ∗ A2 . From the hypotheses A is discrete with λ = λ1 .
Therefore each Ai = (Ai , Pi , ∗, e, logλ i ) is an additive number system and
A = (A, P, ∗, e, logλ ) and A1 are reduced additive number systems.
Notice that A = A1 ∗ A2 since logλ (a1 , a2 ) = logλ a1 1 + logλ a2 2 for all
a1 ∈ A1 and a2 ∈ A2 . Also A1 and A2 have all partition sets with global asymptotic
density since A1 and A2 do.
Consider partition sets of A of the form B = B1 × B2 where Bi is a partition set
of Ai , i = 1, 2.
1 ) = 0. So B
1 (n) ∈ RTλ−α by Lemma 26.
First, suppose Δ1 (B1 ) = 0. Then Δ1 (B
−α1
) and A(n) ∼ A1 (n)A2 (λ−α1 ) by Lemma 17 and
Thus B(n) ∼ B1 (n)B2 (λ
Schur’s Theorem. So, as n → ∞,
−α1
B(log
)
B2 (α1 )
B(n)
λ n)
1 ) B2 (λ
=
= Δ1 (B1 )
→ Δ1 (B
.
−α
A2 (α1 )
A(n)
A(logλ n)
A2 (λ 1 )
Second, suppose Δ1 (B1 ) = 0. The proof that Δ(B) = 0 is exactly as in the proof
B2 (α1 )
.
of Proposition 46. Therefore in both cases Δ(B) = Δ1 (B1 )
A2 (α1 )
Hence for a general partition set B of A
Δ(B) =
k
j=1
Δ1 (B1j )
B2j (α1 )
A2 (α1 )
Asymptotic Density in Combined Number Systems
by Lemma 11 and Lemma 9.4 from [6]. Therefore A has Property I.
81
Proposition 67. If A1 ∗ A2 has Property I and α1 ≥ α2 , then
• α1 = α2 or
• α1 > α2 and A1 is strictly multiplicative or
• α1 > α2 , A1 and A2 are discrete, and λ2 is a power of λ1 .
Proof. Let A = A1 ∗ A2 . Assume α1 > α2 .
Suppose A is strictly multiplicative; then A ∈ RVα by Corollary 9.50 from [6].
Thus A1 ∈ RVα by Proposition 9.55 from [6]. Then A1 is strictly multiplicative;
for otherwise α1 = 0 by Proposition 9.51 from [6] which contradicts α1 > α2 .
Suppose A is discrete; then (a1 , a2 ) is an integer power of λ for ai ∈ Ai . Thus
if there existed an a1 ∈ A1 such that a1 1 was not an integer power of λ then
(a1 , e) = a1 1 would not be an integer power of λ, and likewise for a2 ∈ A2 .
Therefore for all ai ∈ Ai , i = 1, 2, ai i is an integer power of λ; so A1 and A2 are
discrete. Hence λ1 and λ2 are powers of λ, each Ai = (Ai , Pi , ∗, e, logλ i ) is an
additive number system, and A = (A, P, ∗, e, logλ ) is a reduced additive number
system.
As in Proposition 66 A = A1 ∗ A2 and A1 and A2 have all partition sets with
global asymptotic density. So by Proposition 3.20 and Lemma 3.29 from [6], A(n)
∈
−α
RTλ−α ; so A1 (n) ∈ RTλ−α by Lemma 65. Now α = α1 > α2 ≥ 0; so λ
< 1. If
1 (n − 1)/A
1 (n) is infinitely often 1 giving a contradiction.
A1 is not reduced then A
So A1 is reduced, and thus λ = λ1 . Therefore λ2 is a power of λ1 .
Theorem 68. Let A1 and A2 have Property I with α1 ≥ α2 . Then A = A1 ∗ A2
has Property I iff
• α1 = α2 or
• α1 > α2 and A1 is strictly multiplicative or
• α1 > α2 , A1 and A2 are discrete, and λ2 is a power of λ1 .
If A1 ∗ A2 has Property I then for B a partition set of A1 ∗ A2 and for Bij as in
Lemma 11 we get
⎧
k
⎪
⎪
B2j (α1 )
⎪
⎪
if α1 > α2 ,
Δ1 (B1j )
⎪
⎨
A2 (α1 )
j=1
Δ(B) =
k
⎪
⎪
⎪
⎪
Δ1 (B1j )Δ2 (B2j ) if α1 = α2 .
⎪
⎩
j=1
Proof. In the if direction apply Propositions 63, 64, and 66, and in the only if
direction apply Proposition 67.
Because of the connection between discrete multiplicative systems and additive
systems used in Lemma 61 and Proposition 66, global density results for multiplicative systems give global density results for additive systems. Each additive
system can be seen as a multiplicative system (as shown in Section 9.8.1 of [6]);
this translation maps additive number systems in RTρ to multiplicative number
systems in RV− logλ ρ for integer λ > 1 and preserves partition sets and density.
In the next corollary we will also need the fact that if the generating series of the
additive number system diverges at its radius of convergence then the Dirichlet
Karen Yeats
82
series of the corresponding multiplicative number system diverges at its abscissa of
convergence.
For the purposes of this corollary we are returning to the notational conventions
of the first half of the paper.
Corollary 69. Let A1 and A2 be additive number systems such that every partition
set has global asymptotic density. Then every partition set of A1 ∗ A2 has global
asymptotic density iff d1 |d2 or ρ1 = ρ2 .
Proof. Let A = A1 ∗A2 . Pick an integer λ0 > 1. Then Ai = (Ai , Pi , ∗, e, λ0 i ) and
A = (A, P, ∗, e, λ0 ) are discrete multiplicative number systems with A = A1 ∗ A2 .
Apply Theorem 68. Notice that in the case ρ1 < ρ2 the condition d1 |d2 is equivalent
to the condition λ2 is a power of λ1 for the discrete systems.
Corollary 70. Let A1 , . . . , An be multiplicative number systems in RV0 . Then
A1 ∗ · · · ∗ An ∈ RV0 .
Proof. It suffices to prove the theorem for n = 2. Ai ∈ RV0 iff Ai has Property I
and the abscissa of convergence of Ai is 0 by Theorem 10.2 of [6]; the same applies
to A1 ∗ A2 . Now apply Theorem 68.
This gives Odlyzko’s Theorem ([6], 10.5) and the multiplicative half of Theorem
16.1 from [7].
For any multiplicative number system A define Aq , the extension of A by an
indecomposable q, to be the multiplicative number system formed by adding a new
indecomposable q to P, by letting Aq be the set of formal expressions qm ∗ a, for
m a nonnegative integer and a ∈ A, and by extending the norm by the definition
qm ∗ a = qm a.
Corollary 71. Suppose that A has Property I. An extension Aq has Property I iff
A is strictly multiplicative, or A is discrete and q is an integer power of λ, or
α = 0.
Proof. We can form a multiplicative number system Q = (AQ , PQ , ·, 1, ) generated by q by letting PQ = {q}, AQ = {qm : m ≥ 0}, 1 = q0 , qm1 ∗ qm2 = qm1 +m2 ,
and qm = qm . Then Aq ∼
= A ∗ Q, and Q has Property I, αQ = 0. Now apply
Theorem 68.
If A is strictly multiplicative then Corollary 71 gives Theorem 9.69 from [6].
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Department of Pure Mathematics, University of Waterloo, Waterloo Ontario, N2L
3G1, Canada
kayeats@uwaterloo.ca
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